Geometric sum of geometric random variables

I am looking to find the probability mass function of $Y=\sum _{i=1}^{N}{X}_{i}$ where ${X}_{i}\sim \text{Geometric}(a)$ and $N\sim \text{Geometric}(b)$. I attempted to do this by finding the probability generating function of Y and comparing it to known probability generating functions to take advantage of the uniqueness property. (In my searches online, it sounds like I should find that $Y\sim \text{Geometric}(ab)$.)

I am looking to find the probability mass function of $Y=\sum _{i=1}^{N}{X}_{i}$ where ${X}_{i}\sim \text{Geometric}(a)$ and $N\sim \text{Geometric}(b)$. I attempted to do this by finding the probability generating function of Y and comparing it to known probability generating functions to take advantage of the uniqueness property. (In my searches online, it sounds like I should find that $Y\sim \text{Geometric}(ab)$.)